Differential Geometry of Curves and Surfaces by do Carmo Manfredo P.;

Author:do Carmo, Manfredo P.; [Manfredo P. do Carmo]
Language: eng
Format: epub
Publisher: Dover Publications
Published: 2016-09-14T16:00:00+00:00

where a = a(s), b = b(s), c = c(s), s ∈ I . The above formulas are the analogues of Frenet’s formulas for the trihedron T, V, N . To establish the geometrical meaning of the coefficients, prove that

a. c = −〈dN/ds, V 〉; conclude from this that α(I) ⊂ S is a line of curvature if and only if c ≡ 0 (-c is called the geodesic torsion of α; cf. Exercise 19, Sec. 3-2).

b. b is the normal curvature of α(I) ⊂ S at p.

c. a is the geodesic curvature of α(I) ⊂ S at p.

15. Let p0 be a pole of a unit sphere S2 and q, r be two points on the corresponding equator in such a way that the meridians p0q and p0r make an angle θ at p0. Consider a unit vector v tangent to the meridian p0q at p0, and take the parallel transport of v along the closed curve made up by the meridian p0q, the parallel qr, and the meridian rp0 (Fig. 4-21).

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